Normally persimmons are picked in late September trough to October, but these are the kind for drying and they're not picked until much later. Persimmons are not so popular in Australia, but they are grown there mostly for export to Asia. When I lived in Australia we used to go to a grower and buy a box, which was in Autumn there, that is in May. However I think the persimmons in Japan are even more delicious.
BTW in Japanese persimmons are called "kaki" which is also the same sound as the word in Japanese for Oyster, so we often have to specify if we are talking about the fruit or the shellfish.